[See the note at the end of the essay for my account of the importance of these ideas. RC]

My aim in this talk is to attempt to bring about some relationship between learning theory and educational practice. We shall be particularly concerned with the classroom context, with the way in which children develop and how their environment may contribute to or inhibit clear, effective types of thinking.

I should make it quite clear that I am not going to talk about all forms of mental activity. Reverie and fantasy, for example, are forms of mental activity but not necessarily forms of effective thinking. I am going to concern myself with those specific aspects of thinking which are by and large disciplined: such, for example, as reasoning and problem solving, judgement (a somewhat vague concept, since it often means the end result of a process of thought rather than thought itself) ; and finally with so-called `creative' thought -again whatever this may mean.

It seems true to say that disciplined, rational thinking is or should be a major, if not the major, objective of education. Rational thinking will be concerned with the solution of such practical problems as: will this shelf fit into this space? can I afford to buy this object? if I buy this object will I have more value for my money than if I buy that? Effective thinking of this kind will be one of the marks of a well-educated person. It will be concerned too with the solution of personal probems; with the formulation of judgements about people, about politics and morals. As such it will involve emotions and wishes as well as strictly rational elements. Thus one may say that in pursuit of the training of rational thought, an important educational object will be to teach people to distinguish from the strictly objective and rational elements, the emotional and irrational ones which are likely to influence judgement, problem-solving and even tend to determine the choice of problems to be solved and the selection of data to solve them. In addition it should equip us with the techniques for holding these at arm's length.

We can put this in a more concise way and define education as the process of acquiring independence in thought - independence in the sense that the rational process is in fact rational and not unconsciously warped by prejudice. It could indeed be argued that, from the time the child comes into the school to the time he leaves it, his education should be marked by the attainment of a series of increasingly complex intellectual independences. We know for example that the young child passes, given adequate experience, with a certain amount of ease from percepts ,to concepts. We know that the general trend of development is from the concrete to the abstract and that one of the businesses of education is to equip the child with the symbolisms-verbal, numerical, mathematical, musical and so on-which enable him to manipulate abstract thought. Another objective of education might well be stated as that of fostering a move from an absolute rule (whether in arithmetic, the rules of a game, or moral judgement) to the acceptance of the fact that rules are relative and can be modified in the light of circumstances.

Education, too, is I suppose in part at any rate, concerned with the change from emotionally dominated and intuitive (i.e., dependent and partial) to more objective (i.e., independent and rational) thought. At this point we might turn these apparently confident statements into questions and ask ourselves whether in fact these results come about, and if they do, whether this is brought about by education or is simply the consequence of growth and maturation. Piaget and many others have shown us that there appear to be genetic sequences in the thinking of children away, for example, from the sensory and concrete and therefore specific and rigid to more generalized and therefore more flexible types of thinking. However, all along the line there is a triple danger. First, a child may move away from, shall we say, perceptual types of rigidity but in doing so he may develop new kinds of rigidities-for example, the tendency to apply a method of thought or problem solving, blindly; or worse still, he may apply learned solutions regardless of their applicability. At the adult level this is to become the victim of one's experience; in other terms, to fight the war before last. A learned solution is used in inapplicable circumstances and is therefore a wrong and even dangerous one. The second danger is that in teaching - or helping children to learn to think - we may set up interferences by inducing failure and negative emotions, by arousing anxiety and insecurity. This is most clearly seen in what happens in the teaching of arithmetic which is a formal, logical and abstract process with a clearly right and wrong answer. Premature teaching and especially premature abstraction can be severely damaging. But the danger is not confined to arithmetic. It can occur in any aspect of the curriculum, in any educative event of a child's life - wherever in fact he is confronted with demands, expectations and choices he is not mature enough to meet.

The third danger is one of which most of us are unaware but which is pervasive and, in my view, particularly damaging to the intellectually able. In order to help children and adolescents to learn to think, we have, of course, to give them ideas and facts to think about as well as techniques appropriate to the manipulation of ideas and facts; that is to say we tend to have `subjects', organized bodies of knowledge, a definable curriculum. We teach mathematics, physics, history, literature and so on. Thinking and problem solving tend, therefore, to be exercised in relation to specific classes of items of experience. Each area of study has effective strategies of problem solving, types of thinking and so on which are, in part, peculiar to it. One cannot imagine a solely arithmetical technique appropriate to literary understanding. On the other hand, judgement in any field involves, among other things, the definition and delimitation of the problem to be solved, the marshalling of evidence, the determination of its relevance, the setting up of a trial solution and its testing in as objective a way as possible. In teaching subjects-particularly when these are being prepared for examinations-there is a strong and natural tendency to teach them for their own sake, and not to perceive that if generalization of problem solving strategies is to take place, it must be brought about deliberately. There is a good deal of evidence to suggest that one cannot train `faculties' by `disciplines' and that the transfer of training is only of limited automatic occurrence. It is more than open to surmise that if the general elements of problem solving are made apparent to the pupil and if he is taught to apply scientific or mathematical or formally logical forms of reasoning outside the narrow bounds of the subject in which he first learned them, real transfer will take place. One must in short systematically and consciously teach for transfer: otherwise the very effectiveness of our teaching of the specific strategies applicable to the problems of a specific discipline may act as a block to effective thought in many other areas. In illustration, one may remark how frequent it is that scientists fail to apply scientific rigour to emotionally charged problems outside their science and how many `well-educated' people in public and private life are quite incapable of developing problem-solving strategies to meet new problems, but rely upon learned responses whose appropriateness they do not even question.

At this point, perhaps, we might try provisionally to define the dimensions of a problem situation. A problem is perceived as such when the progress to a goal by an obvious route is impossible and when an automatism does not provide an effective answer. At this point one may do one of two things: one may refuse to see that the problem exists, an emotional denial which happens more often than we think, perhaps particularly in committees concerned with education; or one may accept that a problem exists and attempt to define what it is by asking a number of questions aimed 'to understand the propositions that delimit it.

The next stage is to look at the data at disposal and to discriminate those parts of them which are likely to be relevant, to perceive the gaps and, if possible, to fill them with further relevant information. These processes are in one substantial sense imaginative and creative; but a series of even more imaginative and creative processes follows-those of formulating trial hypotheses, of making a selection (if you have them) among learned responses, of developing new responses or of recombining old ones. By and large this will be true, whether you are a child solving a conventional problem in arithmetic or whether you are an adult confronted with a problem in your private life. When some hypotheses and some methods have been formulated, you try them out, you check them; and finally comes the process of evaluation or judgement.

This sketchy analysis makes clear that in the teaching situation (as well as in the ordinary human one) there are a number of qualifying statements which we ought to make. First of all, the person concerned with a problem must perceive - must be motivated to perceive - that the problem exists and he must want to solve it. If he doesn't want to solve the problem, he may not even want to see it and of course a good deal of teaching teaches children that the best thing to do is not to see that a problem exists. Second, it is quite clear that in any problem, even the most elementary, the solver will need to possess or to find relevant items of knowledge. Some of these will derive from his previous learnings and his previous problem solvings; some he may have to search for. Here let me again state the obvious : that a problem for which no data exist cannot be solved; it probably will not even be seen as a problem. This was true, for example, about the `flatness' of the earth; it was true about things like the circulation of the blood. Until certain kinds of information became available in the environment, neither of these was perceived as a problem and no solution could be sought. Thirdly, the solver will need to be master of a good many techniques. For example, if it is a mathematical problem he will need to know how to multiply; if he is going to paint, he will need to know how to mix and apply the paint; and so on. He will need strategies for formulating and testing his hypotheses and he will need some criteria of judgement - probably some form of measurement. These considerations and needs will apply in different ways either at the most abstract level of thinking or in strictly concrete and manipulatory terms.

A little while ago I was watching my very young child - he was about ten months old at the time -playing with the problem of how to open a cupboard door. The door had a little bolt on it which was fairly loose : and this presented him with a series of problems. First of all he pulled the knob on the bolt towards him. When this didn't work, he rattled it and found that by sliding it to the side he could pull open the door. When he wanted to try again, he found that pushing the door back with the bar pushed did not permit him to shut the door. The problem he was seeking did not reappear; there was another in its place. For a long time he played around with this and eventually found out the highly complicated sequence of pulling the bolt back when you'd got the door open and keeping it back if you wanted to shut it. He then went on to repeat this many times, practised it, and sorted it out. This was partly a trial and error type of response and partly highly insightful reasoning. This example illustrates two things; what I have just been trying to state as the dimensions of problem solving; but something much more important, that the problem solving activity of the young child leads to new learning. Even at that level there were certain kinds of techniques which he acquired and which pre-

sumably will make him a menace with any other cupboard in the future. This element of spontaneous acquisition of new techniques and the discovery of new problems is often what is lacking in conventional teaching.

There are many techniques which mankind has developed - diagrammatic representation, verbal symbolism, mathematical and other symbolic languages - as tools for problem solving. We cannot sensibly expect children to discover these for themselves. Hence we have to think in terms of providing circumstances for children in a variety of ways which, as it from the example which I have just given, that trial and error behaviour is characteristic very often of animals, and often of young children. Such behaviour nearly always occurs, even in adults, when the problem cannot be conceptualized. If blind trial and error works, it will with repetition lead to some sort of conditioned response, i.e., learning without insight which may become rigid. If, on the other hand, the trial and error gives way to insight then it will lead to short-cutting which may or may not lead in its turn to a straight conditioned response but at least will give the possibility of transfer and generalization. In terms of learning a specific response, both of these seem to be fairly effective. However, trial and error followed by practice, though fairly effective, is not in all cases either the most economical or educationally, probably, the most desirable way. Moreover if the problem is difficult or impossible to conceptualize and nevertheless the motivations or incentives to solve it are very strong, it may produce a panic type of trial and error - a panic type of reaction-which may in its turn lead either to

inhibition, to avoidance, to loss of interest, to countermotivation, or finally to refusal. This is the sort of thing that you see very often with children who have repeatedly failed in arithmetic.

The educational problem then is not that of avoiding trial and error learning altogether, but of recognizing that it easily degenerates into panic and blockage; and that anyway it may produce an undesirable rigidity in learned responses. What we wish to secure is a situation which permits insight and, through insight, generalization and transfer.

These rather scattered points have a considerable bearing upon the construction of curricula and upon the methods which we use. The first concerns content. To solve almost any problem you need a certain amount of information. There is thus a strong case for the straight learning of facts, relationships, techniques. You cannot dispense with the counters of thinking. Some of these will be provided by the experience given by the child's own environment but many more could be acquired in the course of problem solving and thinking. Here, however, there is a word of warning: for the teacher, knowledge is the result of long past problems and learning and thinking. For the child, the acquisition of knowledge may in fact be either the authoritarian memorizing type of process which equips him with certain facts, solutions and overlearned techniques, but does not necessarily help him to understand the processes underlying them; or, on the other hand, it may be, if it is properly handled, itself a process of thinking. It seems probable that ordinary teaching may, with very similar content, either stimulate the process of thinking itself - the problem-solving which underlies the relationships of facts - avoid, or even deaden it.

How far the problem solving is stimulated or avoided depends on the teacher's own views about how important this is and how important are the relationships involved; but it also depends upon whether the child is or is not mature enough to accept it and to carry through the reasoning processes involved. For example, in teaching a child fractions it may well be found that with certain children at certain stages, it is simpler and more effective to tell them to turn upside down and multiply and encourage them to apply this as a blind technique in the solving of other problems than it would be to get them to understand why you turn upside down and multiply. I am not prepared either to attack or defend this as a procedure. It is possible, and I think all of us in our own educational experience have found this, to learn certain things first in a more or less mechanical way and later only, to develop an understanding of the reasons and principles which underlie them. The danger inherent in this is that it tends to produce a rigid type of response because the child does not understand it. Thus, if for any reason it is necessary to teach in a more or less authoritarian way techniques or facts, then one should also make a mental note to bring about understanding of the principles involved at some later stage.

So much, then, rather sketchily, for the content. More important, I think, is the question of method. Children have to be trained, just as adults do, how to approach and tackle problems. There are certain general notions here as to what should be taught: getting the questions right; making sure that the right question to define the kind of problem is asked; seeking out the relevant data; setting up the kind of hypothesis that can be tested as distinct from the kind of hypothesis which is an untestable noise; the trial or experiment; and the evaluation. These things need to be explicit and are of general bearing; but there are certain other aspects specific to the kind of problem with which the child is confronted. For example, problems involving multiplication or division are quite different from those involving the finding of means to express unique experiences in words, in colour or in line.

These are matched by a third consideration which conditions what is educationally possible and desirable. By and large one risks disaster if children are expected to undertake levels of thinking for which they are not ready. Piaget and others have taught us that there is a progression in the development of conceptualization and in the development of reasoning. An important stage along the road should have taken place before school; children should by five have passed from simple precepts to the preoperational handling of concepts. This however will not necessarily be true over the whole field of child thought, nor for all children. There will be important qualifications related particularly to the social group from which the children come and the kinds of experience their homes have provided. But basically the primary school teacher is concerned with the two stages which Piaget describes as `intuitive thought' and `concrete operations'.

At this point we may quickly refresh our memories of what Piaget calls intuitive thought. He means that intelligent behaviour is limited to overt action and thinking is tied very tightly to perceptual factors; there is no real reversibility and there is very little in the way of conservation. That is, the child can indicate absent objects by the use of signs but they are still concrete in his environment. He does not generalize. This means that real symbolization is for him somewhat abstract and remote. In the next stage, the actions of combining and dissociating, ordering, setting up correspondence, become grouped and are capable of deliberate reversal. This, if you like, is the stage at which the child can manipulate concretely almost anything in his environment but he is still concerned with actual operations on concrete objects. Though he can classify and serialize and can use numbers in a chanting kind of way, they are still very much tied to sensory perceptions, particularly, of course, things like counting and checking and what not. He needs to check his counting and serializing against real perceived models.

Piaget assigns fairly wide age-ranges to these : to the first 4-7; to the second 8-I I years. There is reason, I think, to suppose from some of the English work that these processes will be speeded or slowed according to the.sorts of experience which the child gets but the evidence also suggests that the sequence is none the less fairly fixed, and that, if these sequences are transgressed, there is real danger of inducing kinds of rigidity of thinking which will inhibit learning later on.

The third stage concerns principally the teacher of adolescents. It is the full development of reasoned thinking, of capacity to handle propositional or formal operations. The child is capable of reasoning on a variety of relational factors, and able to accept a proposition `as if it were true' and to reason from it. He has developed operational schemata and the genuine basis of abstract, logical and mathematical thought.

This is a development which does not universally or automatically take place. Nor does it necessarily follow from ordinary educational experience. A great many people in fact do not get beyond the concrete operational sort of stage though they may appear to operate at more abstract levels because of the way they verbalize. In determining whether or not pupils will progress from the concrete operational to a higherstage, the various languages acquired in or improved by education are probably critical-particularly the mother tongue itself, and the mathematical types of symbolism. In the light of this, we may look again at pacing-at the idea of trying to match to the course of a child's development, curriculum and method, the kinds of problems which are put before him and the kinds of strategies which he is helped to use. Two things become immediately apparent: any content, method, or problem which presupposes a stage in development not yet attained is likely to lead the child to a trial and error type of behaviour, and to rote memorizing. If the incentives to learn are authoritarian and punitive, panic and avoidance tend to become marked aspects of behaviour. This is very noticeable, of course, in problems in learning arithmetic and mathematics. Even where the teaching is good and relations between teacher and taught are warm and friendly, premature

teaching may lead to the adoption of rigid strategies without understanding and therefore incapable of generalization. We see this quite often in the child who conforms well to a reasonably humane school and becomes very good at doing mechanical arithmetic but who gets progressively more and more floored as his (more usually her) problems move away from the kind which he or she has learned to solve. The second risk is in some ways more serious. Highly authoritarian teaching may eventually provoke revolt. The accepting, pleasant teacher whom the children very much like makes it difficult for them to let him down. If the environment, that is to say, the home and the school, does not specifically foster the type of growth transition from one state or level of thinking to the next, thus preparing a child for further stages in the content of his curriculum, the growth itself will be delayed and not merely in intellectual ways. One has the not unfamiliar picture of a child who is expected - because of success gained hitherto - to be ready for a further stage and who chronologically ought to be at that stage, but who in fact has become fixated in a form of success which does not allow for growth. They are like a sports player who has acquired a high degree of skill but whose very level of attainment inhibits the recombination and modification of that skill into more complex forms. This, I think, is peculiarly likely to happen to those children whose use of language (in Bernstein's formulation) is in fact largely of the `public type', and who therefore lack the instrument with which steadily to make more and more precise the concepts with which they deal.

As children pass through the educational system, verbal conceptual development becomes increasingly important. It is the first and most flexible series of symbolisms available to thought. This is perhaps one of the reasons why many children who seem to be quite good and quite bright in the early primary years, tend to tail off towards the end of the primary school and decline markedly in the secondary school. Because their language is neither rich nor precise and because the school does not or cannot compensate for this, such children find increasing difficulty in formulating problems of the more complex and abstract type with which they are confronted. They are driven to attempt to reduce the abstract to concrete, manipulatory forms; and if this is impossible, the problem for them cannot be solved. Here, as an aside, we may point out the danger inherent in visual aids. Visual methods which make things simpler by making them more concrete, may tend, if they are not properly used, to fixate children at this stage. Educationally speaking, they are props only and should be deliberately used as aids to the formulation of problems in symbolic forms, particularly perhaps in the form of verbal symbolization. Otherwise the visual aid, while appearing to simplify the teaching task, may in fact contribute to inhibit subsequent development.

In the light of what has so far been said, I should like to consider some methods and to draw your attention to an excellent article, `Teaching Problem Solving', by Williams in Educational Research. Williams takes the evidence for various methods of helping children to learn effective thinking and discusses this in the light of what can be done in the classroom. He points out, for example, that there is considerable evidence that with young children concrete presentation of problems is effective but that with adults and, particularly with intelligent adults, it is not as effective. This is what you would expect from the work of Piaget and others; that as the more effective tool of symbolization becomes dominant in thinking so the merely sensory (concrete) presentation of a problem in specific form is likely to be less efficient, less general and less effective. With primary school children, however, it is possible and necessary to ease this transition. Well designed structural material for the teaching of arithmetic and mathematics, for example, because it varies the way in which problems are presented, provokes different, strategies for solving them. If the teacher knows what he is about, he can use this material to encourage the child to conceptualize, to ease the transition to symbolic manipulations, and to induce the development of increasingly generalized types of attack.

It has been shown, in a number of experiments, that children - even quite young children - are greatly helped if they verbalize as they look at the concrete presentation of their problem. This seems to be because verbalization helps to designate, to discriminate and to classify. Children remember objects better if they name them than if they simply put a ring round the word designating them; they will learn better if they accompany their learning as they go with verbal formulations of what they acquire; they will solve problems better if they talk aloud while they are doing them. This suggests that it is likely to be helpful (in many cases, though not perhaps in all) to stimulate discussion of the problem by the children before they attempt to solve it. With older children it is worth while asking them to formulate on paper an analysis of the problem and to set up a plan of attack. It can also be helpful to the teacher since, if the child fails to formulate the problem clearly and correctly, diagnosis of his difficulty is much easier. Katona and others have shown that diagrammatic or graphical representation will also help in the solution of certain types of problems. Williams gives a good example, here, which is quoted because it illustrates how certain types of diagrammatic presentation will help. The problem is of the familiar type : `John gave Joe twice as much as Jim gave George' (the alliteration is there, of course, as a distractor) ; `Joe gave George half of what he received; if

Jim gave George sixpence, how much did George receive altogether?' If this is dealt with diagrammatically, many people will find it easily solvable who might otherwise have found it difficult.

This calls, of course, for a simple type of diagrammatic representation and is not a unique way of attack; but there are other problems which can be solved only in this way. The research work on problem solving suggests that there are individual differences in the usefulness of diagrammatic techniques and, of course, differences inherent in the types of problem. In any case, however, it nearly always helps, with a complicated problem, to list the items of information in some patterned way and try these patterns out. No method is likely to work with all problems, with all children and every time.

Finally, a good deal of work suggests that group methods of problem solving may also be useful. The advantage of this form of attack is that many more hypotheses tend to be thrown up by a group than one person would be likely to think of: groups are more productive of hypotheses and therefore are likely to be more productive of solutions than single persons, though in fact they take more time. The solutions reached tend to have a higher quality; a matter of importance when you are confronted by problems which have no right, absolute and unique solution. There tends also to be a higher level of criticism of the hypotheses and of the solutions in a group. This technique has obvious educational value, especially in teaching children how to solve more complex types of problem, particularly those which do not have the inherent simplicity of mathematics but are more `real' in the sense that they touch the untidiness of life. Attention should, however, be drawn to another important and educa-

tionally valuable aspect. Within a group situation, it is possible to encourage the separation of the hypothesis forming from the evaluating type of activity. What one so often finds with children is that they have become so critical of their own powers that they are inhibited from producing a large number of hypotheses - good and bad - from which to choose. This may (and often does) mean they are inhibited from producing the apparently bizarre hypothesis which ultimately may be the right one. If a group is stimulated to throw everything into the ring as it were and to exercise critical choice only at a second stage, while much rubbish will be produced, much that is creative is enabled to reach the surface. The tasks of creation and criticism can be separated and trained separately with advantage.

This draws attention to a principal impediment to problem solving-anxiety. Anxiety seems to impair performance more in complex than in simple tasks. The fear of failure inhibits search and may prevent it altogether; over-confidence on the other hand tends to too facile types of acceptance. The problem, educationally, is that of how a child's morale can be maintained so that he will go through the stage of producing all sorts of hypotheses if these are required without feeling too critical about them and at a second stage he can be brought to exercise discrimination and judgement. If children are expected not to produce hypotheses which seem to be silly and if we criticize those which seem to be wild, we may in fact inhibit the more creative type of thought. This is one of the difficulties in the teaching of arithmetic. It does not permit of. the bizarre type of hypothesis. The difficulty in teaching something like English is that it does, but that it is difficult for children to have any means of judgement that is adequate. Between these two extremes lies the whole sequence of problems in the teaching of thinking.

We also know that if the problem is too difficult, the ensuing frustration will tend to constrictiveness and to reduction in fluency. The child will just cease to try. This means, obviously, that one tries to avoid the over-difficult type of problem; even more important is that one should make sure that the child confronted with a problem does have some sort of procedures which will enable him to get to grips with it. The understanding of techniques or procedures can be of two kinds: he may have a sufficient mastery of an appropriate technique in the sense that he can apply it; or he may understand the principles which underlie it. In the first case, provided the problem and technique fit, all will be well; but if there is an essential variation in the nature of the problem, then some understanding of underlying principles will be necessary to adapt the technique and make it serviceable.

This suggests that - even if it takes longer - children should be taught the rationale of the techniques they use and should experience wide differences in their application. Again, however, we must be neither simple minded nor absolute. Research seems to confirm common sense when it points out that, if you are dealing with intelligent children then it certainly pays to teach them the principles; but that if you are teaching duller children and over a fairly short term, it probably does not. A third proviso also turns up. If the generalization of the principles underlying a technique is too limited, this in itself may inhibit the intelligent child quite strongly. Finally, for our somewhat dubious comfort, it should be said that many intelligent children survive the wrong kind of teaching and develop wide and flexible strategies for themselves. This might lead us seriously to ask the question-relevant especially for opponents of any form of streaming, even of groups or sets within classes - whether it might be wise, in a class of widely varied ability, to teach all children problem-solving technique routines, in the hope that the dull ones would have something to get on with and the bright ones might be able to survive and surmount the routine. This, I suspect, would be a counsel of despair but it certainly looks as though some serious thought should be given to it by advocates of ultimate comprehensiveness.

Within any subject area we tend to teach the techniques of thinking, reasoning, problem solving, specifically appropriate to that area and we ignore or do not realize how far the techniques of one discipline, with slight or major alterations, are applicable elsewhere. Here again, it looks like educational sense to provide a wide variety of problems from different areas in the hope that, through the variety of techniques they are forced to apply, children will learn the common principles which underlie them. However, we must enter a caveat. If you have plenty of time, then it is probably better in the long run to proceed with plenty of variety. If time is short - if the examination is a fortnight away - then it is probably better and more economical to proceed with quite specific, more or less authoritarian, techniques for olving problems. The root of this is simple and commonplace enough; in using any particular technique a certainamount of practice is necessary; where you provide a variety of problems you of course tend to diminish the total amount of practice given on any one technique. The result, unless great care is exercised, is a width of partial mastery at the expense of real control though of many fewer techniques.

We are led to suggest to any teacher with any class that he makes clear to himself both the immediate and the ultimate aims he wishes to achieve in problem solving and that he carefully controls and records the length and type of practice which each child gets. This dazzling (but often ignored) glimpse of the obvious brings us to the question of whether children should discover or whether we should guide or even transmit discovery. There is a lot of educational cant about this. Some enthusiasts seem to say that children should be allowed to discover any old thing in any old way and we should hope for the best. This is tantamount to suggesting that in ten years, children will find out what it has taken, I suppose, several thousands of mankind to find out over centuries. At the other extreme are those who wish to instruct their pupils in everything which it is considered they should know and who believe that what sticks will be an intellectual capital for a life time. It is quite clear that genuine and effortless discovery leads children (and adults) to reorganize their information in newer and meaningful ways; that, in dealing with one problem situation successfully, they are

likely to transfer what they more or less painfully acquire to similar new problems. But unguided discovery may lead even intelligent children to discovering inefficient techniques; and the partial success stamps these in. This again is like the tennis player who teaches himself and can never get beyond a certain level because the skills he has, as it were, overpractised inhibit the learning of higher level or more effective ones. Once more research confirms common sense. Discovery by itself is likely to be much less useful educationally than discovery plus guidance. Practically, this seems to mean that the teacher does not tell the child how to solve the problem but that he asks leading questions, he gives hints, he arranges the situation so that the child in fact tumbles into discovery by an effective route. Cross-evidence, however, suggests caution. It seems that a permissive atmosphere tends to favour the intelligent and a more prescriptive and directive one the less intelligent. This is in slight contradiction of some modern philosophies of education and I am not going to take sides, except to say that there are degrees of permissiveness. Practically, it is sometimes necessary, with not very bright children, to give them the security of a good deal of directiveness and then to proceed by questioning and hinting. How far this is done is a matter for the sensitivity of the teacher to the level at which the child is and how far he is motivated to tumble over into a solution.

So far little has been said about the way in which emotion may directly falsify reasoning and problem solving - a matter of the utmost importance educationally. It seems to me evident that we have to do something in education about the impact of prejudice and rigidity, and about emotionally determined blocks on reasoning. Here the general suggestion may be made that we should see to it that education confronts pupils not merely with variety within the type of problem but with a variety of kinds and areas of problems from the unique solution type we find in mathematics to the compromise solution type which in fact is characteristic of almost every kind of problem outside mathematics. There are in fact few human problems which have a uniquely determinable solution; and it is important that the real problems of the classroom - disciplinary problems, personal problems - and as you get on into adolescence, ethical and other kinds of problems - should be attacked in a problem solving and not in a didactic kind of way. Secondly-and this too is particularly true at adolescence-it is important to make the technique for solving all kinds of problem explicit and to show that these techniques can be transferred : that the kind of mental process which goes on in looking at a mathematical problem is in fact generically similar to the more complex processes and strategies demanded by personal and social problems. It is worth while, for example, to get children to write down the pro's and con's of some problem which vexes them. It is worth while trying to get them to set up, on this basis, a series of hypotheses, in imagination, in real situations. It is worth while indicating to them that there are differing degrees of rigour in the way in which people use language and that very often words which seem to be just words conveying items of information or facts, may be heavily emotionally charged. `Race' may be taken as an obvious example to show how a word may have a meaning apart from its intellectual content and how, in any discussion, this emotional content will tend to interfere with objective thought.

While language is perhaps the most evident area in which this occurs, it can occur in other forms of symbolization. In this connection it is worth while giving practice in the transposition of statements and reasoning on problems charged with emotion from verbal to other symbolisms; and similarly that of turning visual material into words. I once came across an interesting piece of research, which I have never succeeded in finding again, in which propositions were put in two ways : in the first, statements were put in symbolic form - all A are B, therefore all B are A, with a request to mark them true or false. Then, in the second part of the test, statements of similar logical form were expressed in words with highly charged emotional contents-for example, 'all communists are trade unionists, therefore all trade unionists are communists'. It was found, even with quite intelligent university graduates, that the interference of the emotionally charged presentation was marked. They were American university students, of course, and I am sure that this wouldnot happen here: we never see any examples of this in the press or anywhere else!

Finally, it is worth while to draw attention to the fact that any hypothesis, if it is to be something more than a pious opinion, must be put in a form that permits verification. Vague definitions, rolling phrases ('good citizens', `high standards of moral behaviour') have to be made precise and put in operational terms before they can be verified. In this, in most of the areas of life that count, prejudice and emotion tend to colour and distort. Hence perhaps the most critical task of all for education is that of training pupils to reflect upon their own thought processes and particularly upon the way in which prejudices may colour their choice of problems; the kinds of data which they are prepared to admit as part of the solution; and how such factors influence the evaluation of the solution once they have arrived at it. This seems to me to be eminently a job for the teacher of adolescents, but it is not solely his. Young children suffer from emotional distortions - and so, of course, do their teachers.

xxxXXXxxx

Explanatory note, in the year 2011. I read this in 1966 as an undergraduate at the University of Tasmania. On a (now-faded and worn) card I recorded the following notes:

1. Accept that a problem exists and attempt to define what it is by asking a number of questions aimed to understand the propositions that delimit it.

2. Look at the data, discriminate relevant parts and if possible fill in gaps.

This introduction to the problem-oriented mode of procedure was probably enabled me to pick up the central theme of Popper's thinking when I encountered it later. One of his books is called All Life is Problem-Solving!